Timeline for on Brieskorn Manifolds
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2013 at 9:27 | comment | added | Francesco Polizzi | $M(3,2)$ is homeomorphic but not diffeomorphic to $\mathbb{C}$. The inverse of the normalization map $\mathbb{C} \to M(3,2)$ given by $z \to (-z^2, z^3)$ is continuous but not differentiable at $(0, 0)$. As S. Carnahan correctly said, by the Proposition in Milnor's book there can be no smooth manifold underlying the analytic structure of $M(3,2)$. | |
| Nov 1, 2013 at 11:29 | comment | added | S. Carnahan♦ | @sife I suspect he means that the analytification of $M(3,2)$ does not have a smooth manifold underlying its analytic space structure. | |
| Nov 1, 2013 at 1:26 | comment | added | sife | How to understand that "A complex variety can never be a smooth manifold throughout a neighborhood of a singular point."? Because the topological space $M(3,2)$ admits a smooth structure. In other words, "the complex variety $M(a_1, \ldots, a_n)$ is never a smooth manifold if $a_i \geq 2$" is not true. | |
| Oct 31, 2013 at 18:25 | comment | added | Francesco Polizzi | Yes, $M(3,2)$ is the affine cuspidal cubic curve and the map you give is the normalization $\mathbb{C} \to M(3,2)$. | |
| Oct 31, 2013 at 17:25 | comment | added | sife | $M(3,2)$ is homeomorphic to $\mathbb{C}$. This can be seen by $z\mapsto (z_{1},z_{2})=(-z^{2},z^{3})$. So $M(3,2)$ admits a smooth structure. | |
| Oct 31, 2013 at 16:04 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Oct 31, 2013 at 15:59 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Oct 31, 2013 at 15:57 | comment | added | Francesco Polizzi | right, the Milnor links. | |
| Oct 31, 2013 at 15:54 | comment | added | sife | $M(3,\, 6r+1, \, 2, \ldots, 2)\cap S^{4k+1}_{\epsilon} \quad \textrm{and} \quad M(3, \, 2, \ldots, 2)\cap S^{4k-1}_{\epsilon}$ are homotophy spheres of dimension $4k$ and $4k-2$, respectively. | |
| Oct 31, 2013 at 14:53 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Oct 31, 2013 at 14:45 | comment | added | Francesco Polizzi | I do not know if the complete answer is known. I edited the answer with some examples of Brieskorn varieties that are homotopy spheres. | |
| Oct 31, 2013 at 12:15 | comment | added | sife | When is $M(a_1, \dots, a_n)$ a topological manifold? | |
| Oct 31, 2013 at 12:05 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Oct 31, 2013 at 11:51 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Oct 31, 2013 at 11:44 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |